All Questions
Tagged with reference-request dg.differential-geometry
800 questions
4
votes
0
answers
552
views
Lattices of $\mathbb{R}^s \ltimes_\varphi \mathbb{R}^k$
Edit: Thoughts updated (22/3/2021).
I've come across with the following problem.
Let $G=\mathbb{R}^s \ltimes_\varphi \mathbb{R}^k$ where $\varphi:\mathbb{R}^s\to \mathrm{Aut}(\mathbb{R}^k)=\mathsf{GL}(...
7
votes
0
answers
248
views
Does the Hodge decomposition hold for equivariant differential forms?
Let $M$ be a Riemannian manifold. The Hodge decomposition tells that
$$
\Omega^*(M) = \mathrm{im} \ d \oplus \mathrm{im} \ d^* \oplus \mathscr H^*(M)
$$
where $d^*$ is the adjoint operator of the ...
- 2,789
5
votes
0
answers
324
views
Earliest reference for infinitesimal neighborhoods of the diagonal
Where was $I_x/I_x^2$ first introduced? (DG or AG) asks about the algebraic cotangent space. The paper First neighborhood of the diagonal and geometric distributions by Kock claims Grothendieck ...
- 10.5k
6
votes
1
answer
160
views
Metrics of non-negative sectional curvature on $S^7$-bundles over $S^8$
In Curvature and symmetry of Milnor spheres, Grove and Ziller construct metrics of non-negative sectional curvature on $S^3$-bundles over $S^4$ (by using a cohomogeneity one action). In the same paper,...
- 641
15
votes
6
answers
2k
views
Any shortcuts to understanding the properties of the Riemannian manifolds which are used in the books on algebraic topology
I'm now attending a reading seminar on the algebraic topology.
The seminar treats the book of Bott & Tu (Differential Forms in Algebraic Topology) and Milnor (Characteristic Classes).
In those ...
- 1,013
15
votes
3
answers
875
views
Laplacian on manifolds and random matrix theory
Let $M$ be a compact Riemannian manifold with a metric $g$, and consider the spectrum of the Laplacian operator $\Delta$.
What is known about the relationship between this spectrum and random matrix ...
- 2,087
10
votes
1
answer
3k
views
Taylor expansion of the metric tensor in the normal coordinates
I am looking for a reference with a Taylor expansion of the metric tensor in the normal coordinates.
The coefficients should be written in terms of $\mathrm{Rm}, \nabla\mathrm{Rm}, \nabla^2\mathrm{Rm},...
- 45k
1
vote
0
answers
213
views
Injectivity radius bounds for Riemannian manifolds of low regularity
In their seminal paper, Jeff Cheeger, Mikhail Gromov, and Michael Taylor derivated bounds on the injectivity radius of Riemannian manifolds with bounded sectional curvature of the form:
$
inj(p)\geq r ...
7
votes
1
answer
448
views
What would be a good introductory reference for learning jet-bundle theory?
I am interested in learning the theory of Jet bundles, and am aware of the standard reference "The geometry of jet bundles" by D. J. Saunders. However this appears to be a detailed book, ...
- 709
3
votes
1
answer
190
views
Laplace eigenfunction on a polygonal domain symmetric about an axis
Consider a polygon $\Omega \subseteq \mathbb{R}^2$, and let us consider the usual Laplacian operator $\Delta = \partial_x^2 + \partial_y^2$, with Dirichlet boundary conditions. My question comes from ...
- 33
2
votes
0
answers
141
views
For a 1-parameter family of metrics, how do we compute the derivative of the intrinsic geometrical objects like curvature, Hessian, etc
Consider a family of metrics and functions $(g(t), u(t))$ on $M:= \mathbb{R}^3 \setminus B_1$ satisfying
$$ g(0) = g_0, \quad g'(0) = \tilde g, \quad u(0) = u_0, \quad u'(0) = \tilde u$$
where $g_0$, $...
- 969
5
votes
1
answer
176
views
Reference for local linearization theorem
I would need to reference the following seemingly very well known fact:
If f:$M\to M$ is a diffeomorphism of finite order, then at any point in the fixed-point set of f the manifold M has coordinates ...
- 183
5
votes
1
answer
564
views
Geometric invariants of a Riemannian manifold encoded in certain moment map
Let $(M,g)$ be a Riemannian manifold with isometric group $G=Iso(M,G)$. The metric gives an isomorphism between tangent and cotangent bundle of $M$. So $g$ induce a natural symplectic structure on $...
- 356
4
votes
0
answers
236
views
Fréchet subdifferentiation on riemannian manifolds
Context. I'm looking for a "natural" definition of subdifferentials on riemannian manifolds.
Given a function $F:\mathbb R^m \to \mathbb R$, its Fréchet-subdifferential at a point $w \in \...
- 6,853
4
votes
1
answer
503
views
singular metric (with essential singularity)
Working on some $Q$-curvature equation in dimension $4$, I have been faced with singular metric of the form $(\mathbb{B}, e^{-1/\vert x\vert ^2} \vert dx\vert)$. I try to figure out to what those ...
- 914
1
vote
0
answers
328
views
Codifferential of wedge of two 1-forms
Let $\omega,\eta$ be two 1-forms on a manifold $M$. I'm interested in an expression for
$$
\delta(\omega\wedge\eta)
$$
where $\delta$ is the co-differential operator $\Lambda^2(M)\to\Lambda^1(M)$. ...
- 213
1
vote
0
answers
153
views
Reference request for a paper of Berard-Bergery
I was wondering if anyone could point me to a pdf copy of the following paper by Lionel Berard-Bergery:
"Scalar curvature and isometry group", in Spectra of Riemannian Manifolds, Kaigai ...
- 1,903
1
vote
0
answers
55
views
Projection of a real analytic manifold onto subspace is union of real analytic submanifolds
Let $M$ be a compact connected real analytic submanifold of the Euclidean space $\mathbb{R}^{n} \times \mathbb{R}$ and denote by $\pi : \mathbb{R}^{n} \times \mathbb{R} \rightarrow \mathbb{R}^{n}$ the ...
- 19
7
votes
2
answers
358
views
Cone unfolding of space curves
There is a natural length-preserving operation which transforms any rectifiable space curve $\gamma\colon [a,b]\to R^n$ into a planar curve $\tilde\gamma \colon [a,b]\to R^2$. This operation, which ...
- 7,149
8
votes
1
answer
856
views
Are there mistakes in Kovalev's "Twisted connected sums and special Riemannian holonomy"?
This is kind of a strange and vague question... sorry about that.
I am really interested in $G_2$ Twisted Connected sums as described in this paper: https://arxiv.org/abs/math/0012189 "Twisted ...
user117832
2
votes
0
answers
269
views
Solvability of a PDE involving the Dirichlet-to-Neumann operator
Let $M = \mathbb{R}^3 \setminus B_1$ where $B_1$ is the unit ball (equip $M$ with the euclidean metric for simplicity, but it will be replaced by an arbitrary asymptotically flat metric).
Let $N: L^2(\...
- 969
6
votes
2
answers
448
views
About the index theorems
I am looking for some introductory book/paper/notes about the several index theorems and their applications. By several I mean the "classical" Atiyah-Singer theorem, the local index theorem (...
- 789
2
votes
0
answers
125
views
Variation of Morse Functions: a reference request
Suppose I have a manifold $X$ and a family of Morse functions $F_t:X \times \mathbb R \to \mathbb R$ on it where $t$ is the second parameter. So, if we fix $t$, we get a regular Morse function for ...
- 7,746
3
votes
0
answers
247
views
Fibre metrics on non-linear bundles
Usually what is meant under a fibre metric is that one is given a (smooth) vector bundle $\pi:Y\rightarrow X$, and on each fibre $Y_x$ an algebraic inner product $g_x$ that varies smoothly from point ...
- 2,792
1
vote
0
answers
77
views
Stratification of the space of maps transverse to another given one
If $M,N,S$ are manifolds ($M$ and $S$ compact) and $g:S\to N$ is a smooth map, then if we endow $C^\infty(M,N)$ with the strong topology we get that
$$\pitchfork(M,N;g):=\{f\in C^\infty(M,N)\,;\,f\...
- 31
15
votes
2
answers
2k
views
Reference request: the theory of currents
I am a graduate student and want to study the theory of currents. What is a good reference for a beginner?
I should be familiar with the theory of distributions or generalized functions on $\mathbb R^...
- 151
10
votes
0
answers
426
views
Gromov's compactness theorem via Sacks-Uhlenbeck and Schoen-Uhlenbeck
Gromov's compactness theorem for pseudo-holomorphic curves, in section 1.5 of "Pseudoholomorphic curves in symplectic manifolds," is very well known. I'm aware of the following two proofs:
...
- 2,247
2
votes
3
answers
336
views
For globally conformally flat surfaces, does radial symmetry of conformal factor imply the surface is a sphere?
We know that for the unit sphere in $\mathbb{R}^3$, the standard metric on such a sphere can be written as $$g=\frac{4}{(1+x_1^2+x_2^2)^2}(dx_1^2+dx_2^2)$$by a stereographic projection map from the ...
- 1,350
2
votes
0
answers
152
views
Period map on non-Kähler manifold
Is there a theory of period map on non-Kähler manifolds that has Hodge decomposition? Any reference is helpful. Thank you.
- 263
4
votes
0
answers
98
views
Spectrum of Laplace-Beltrami with piecewise constant coefficients
By the Laplace-Beltrami with piecewise constant coefficients I means the operator $-div (f\, \nabla .)$ in the 2-sphere. Where $f$ is a piecewise constant function that takes two values $1$ and $a>...
- 61
2
votes
2
answers
452
views
Reference for the divergence theorem for embedded $C^1$-submanifolds of $\mathbb R^d$ with boundary
I'm aware of Gauss's theorem (aka the divergence theorem) for compact subsets $K$ of $\mathbb R^d$ with "$C^1$-boundary"$^1$.
I know that there are several generalizations of this theorem, ...
- 167
0
votes
0
answers
126
views
mean curvature for codimension $>1$?
The mean curvature of a hypersurface in a Riemannian manifold is defined to be the trace of the second fundamental form. I was curious, does the notion of mean curvature generalise to higher ...
- 3,625
6
votes
1
answer
439
views
Name for a class of almost symplectic manifolds
A $2n$-dimensional manifold $M$ is said to be almost symplectic if it possesses a non-degenerate two-form $\omega \in \Omega^2(M)$. Equivalently, an almost symplectic structure is a $G$-subbundle $P \...
- 33.3k
5
votes
1
answer
147
views
Equivalence generated by Jacobian minors
Let $f,g:\mathbb{R}^m \to \mathbb{R}^n$ be two smooth functions and let $k$ be a strictly positive integer. Write $f \sim_k g$ if at each point in the domain, the determinants of all $k \times k$ ...
- 15.5k
6
votes
1
answer
229
views
Does $\pi_k(M)\neq 0$ implies $\operatorname{ind}(\gamma) < k$?
Cross post from MSE. and sorry if this is an obvious question.
Here is a line of proof of Theorem 1.15 from
Brendle, Simon, Ricci flow and the sphere theorem, Graduate Studies in Mathematics 111. ...
- 4,195
7
votes
0
answers
265
views
Generalized differential geometry based on Penrose's abstract tensor systems?
Penrose graphical notation has been an important precursor of string diagrams for monoidal categories. It was introduced in Penrose's paper Applications of negative dimensional tensors with intended ...
- 6,406
0
votes
0
answers
84
views
Prerequisites/Preparation for understanding a research paper - global solutions to Einstein field in Bondi Coordinates
I would like to read this paper:
João L. Costa, Filipe C. Mena, Global solutions to the spherically symmetric Einstein-scalar field system with a positive cosmological constant in Bondi ...
- 101
1
vote
0
answers
137
views
Invariant subspace of a nonlinear map
First please see this very simple fact:
Fact: $\ $ Any linear map $T: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ has a proper invariant linear subspace.
By an invariant subspace we mean a space $M$ ...
- 111
18
votes
1
answer
566
views
Subgroup $\mathrm{E}_6$ generated by $\mathrm{Spin_7}$ and $\mathrm{SL}_3$
Let $\mathbb{O}$ be the octonion algebra (say over $\mathbb{R}$) and let $J_{3}(\mathbb{O})$ be the set of $3 \times 3$ hermitian matrices with octonion coefficients, that is:
$$ J_3(\mathbb{O}) = \...
- 7,300
2
votes
1
answer
381
views
Question about Berezin line bundle of odd cotangent bundle of supermanifold $\text{Ber}(\Pi T^*\mathcal N)$
The followings are from Mnev's paper about BV formalism.
Example 4.15 (Definition of split supermanifold)
Let $E \to M$ be a rank $m$ vector bundle over $n$-manifold $M$, then there exists a ...
- 79
2
votes
0
answers
67
views
Closed-form expression for Riemannian exponential maps on symmetric spaces
Besides the Poincaré model for the hyperbolic disc, $S^n$, and Euclidean space, what are known instances of a symmetric space $M$ (finite-dimensional) for which the exponential map is known in closed-...
- 5,405
1
vote
1
answer
93
views
Initial value problems on manifolds around submanifolds (reference)
I am looking for a reference on initial value problems formulated on smooth manifolds with initial conditions on submanifolds. More precisely, let $X$ be a smooth manifold and $Y\subset X$ a embedded ...
- 11
3
votes
1
answer
280
views
About the metric and embedding of sphere
Let $S^2$ be the $2$-dimensional sphere with a metric $g$.
Q:
Can we or how to find a smooth map $f:S^2\to \mathbb R^3$, such that
(1) $f$ is diffeomorphic to its image $Im(g)=:M$,
(2) $M$ ...
- 1,915
1
vote
0
answers
61
views
Splitting formulas for spectral flows
I'm asking if there are splitting formulas for equivariant spectral flow and higher spectral flow (of Dai-Zhang) for paths of Dirac operators, concerning gluing together two smooth compact Spinc ...
- 11
4
votes
0
answers
136
views
Geometrical proof of Noether Theorem [duplicate]
I am reading a very nice Physics book "The standard model in a nutshell" by D.Goldberg and just read there a mention to Noether Theorem. Of course I knew this outstanding theorem very well from ...
- 4,361
2
votes
0
answers
172
views
Construction of Kahler Einstein Metric of Poincare Type
I am reading Kobayashi's Kahler-Einstein metric on an open algebraic manifold. In this paper he constructs a Kahler-Einstein of Poincare type on an open manifold X' = X\D, where X is projective and D ...
- 31
1
vote
0
answers
90
views
Is there any name/occurence to this sequence of numbers?
I am curious if there is any name for this sequence of numbers, or any occasion that this sequence is used.
The sequence is $(c_1,c_2,c_3,\cdots)$ with recursive formula
$$c_n=\frac{1}{2n+1}\sum_{i=...
- 323
15
votes
2
answers
2k
views
Riemannian manifold as a metric space
I am looking for a reference to the following simple statement; it must be classical. (It is easy to proof, but I want to have a reference.)
A metric space $X$ that corresponds to a Riemannian ...
- 45k
6
votes
2
answers
317
views
Quasi-isometric embedding of graphs in non-compact riemannian surfaces
Given a complete riemannian surface $(S,m)$, where $S$ is homeomorphic to $\mathbb{R}^2$, I would like to find a weighted graph $G$ (which means a graph with real non-negative weights on the edges), ...
- 935
2
votes
1
answer
210
views
CW-structure induced by Morse function on Riemannian manifold [duplicate]
I have heard a statement in the following direction. Given a compact Riemannian manifold $M$ with a Morse function on it possibly satisfying some extra assumptions. Then this data induces a CW-...
- 21.8k